Locales as geometric objects

Question

There is the following analogy:

$$\begin{array}{cc} \text{frames} & - & \text{commutative rings} \\ | && | \\\text{locales} & - & \text{affines schemes}\end{array}$$

Here, $\bigvee$ is analagous to $\sum$ and $\bigwedge$ is analogous to $\prod$. The category of locales is defined as the dual of the category of frames. We may also define the category of affines schemes as the dual of the category of commutative rings, this is done for example by Toen and Vaquié in relative algebraic geometry. But usually we define affine schemes geometrically as locally ringed spaces. Also, affine schemes are the local building blocks for schemes.

This leads to the following two questions:

1. Is there a geometric definition of locales?
2. Are locales the local building blocks for some geometric objects?

So, the second question is about completing the following analogy:

$$\begin{array}{cc} \text{frames} & - & \text{commutative rings} \\ | && | \\\text{locales} & - & \text{affines schemes} \\ | && | \\ \text{?} & - & \text{schemes}\end{array}$$

After skimming through the nlab article on locales, I would guess that locales may be defined "geometrically" as localic toposes and that perhaps the glued geometric objects are toposes, too (which ones?). This is motivated by the fact that we have a fully faithful left adjoint functor $$\mathrm{Sh} : \mathsf{Loc} \to \mathsf{Topos}$$ from locales to toposes given by taking sheaves, and that toposes are sometimes regarded as geometric objects. (It is noted at the nlab that locales coincide with Grothendieck $(0,1)$-toposes. But this is just a reformulation of the definitions.)

But this doesn't really look similar to the construction of the spectrum of a commutative ring. It rather looks like the fully faithful embedding from (affine) schemes into the category of cocomplete symmetric monoidal categories, given by taking quasi-coherent sheaves. And this is not how we usually define affine schemes geometrically.

Perhaps there is an embedding from the category of locales into the category of locally ringed spaces? Of course, we cannot take the associated topological space of points, since that may turn out to be empty. Perhaps there is a different construction, though.

Answer 1

First, if you haven't already you should have a look at this introductory paper by P.T. Johnstone The Art of pointless thinking which gives a lot of insight on how locale theory works.

Here are some observations which I hope will answer your questions:

1. As I said in the comment, when you glue locales together along open subspaces in a way similar to affine schemes, the objects you get aren't new, they are just bigger locales so there is no need for new objects (the missing corner is "locales" again).

2. In this picture, toposes are more like Stack/groupoid objects. The analogy is not perfect; more precisely toposes are "kinds of stacks" but do not correspond to those one consider in algebraic geometry like the Artin stacks or the Deligne Munford stacks. This point of view on toposes is I think most visible in Marta Bunge An application of descent to a classification theorem for toposes (I haven't found a freely available version) but also appears in some Paper by I. Moerdijk and of course all of this is a consequence of the amazing and famous Joyal & Tierney "An Extension of the Galois Theory of Grothendieck" (which is unfortunately also not simple to find freely available). In this sense locales are the building blocks for toposes.

3. Locales are indeed a special case of toposes; they corresponds to the "localic toposes" which are exactly the toposes that are generated by subobjects of the terminal object. This notion of localic topos can be promoted to a notion of localic morphism and is rather important in topos theory (basic theory of this notion can be found in A.4.6. of P.T. Johnstone's Sketches of an Elephant).

(This paragraph is not meant to be formal.) To some extent the idea of topos is an extension of the idea of locale: in topos theory you are trying to do topology not with open subsets as the basic objects, but with sheaves as the basic objects. A sheaf (of sets) on a space is always obtained by gluing open subspaces together (along open subspaces) so in some sense sheaves are a generalization of open subspaces. From this perspective locales are just the toposes whose "topology" can be generated by open subspaces (the use of the word topology here is informal and has not much to do with Grothendieck topology).

4. To come back to the question of seeing locales a some sort of geometric objects. As I already said in the comment, such a general point of view of people working with locales is that locales ARE geometric objects by themselves, somehow more fundamental than topological spaces. It's not locales that are "structured topological spaces" -- it's topological spaces that are structured locales (a topological space is the data of a set of points $X$, a locale $L$, and a surjective map of locales from $X$ to $L$, so they are just locales with a specific set of points marked).

5. From the discussion in the comments I get that what troubles you is to consider as geometric an object which might have not any points; I will try to address that in the end of my answer.

First, as you probably know, there is an equivalence of categories between locales having enough points (it mean enough points to distinguish the elements of the frame) and sober topological spaces. So in a first approximation one can consider that locales without (enough) points and non-sober topological spaces are pathological objects, and that except for those pathologies, locales and topological spaces are the exact same things. That is what people were doing for quite some time (the first example of point-free locales/toposes were considered as being completely pathological objects).

It appears that there is a lot of very interesting example of locales without points and that those are not that pathological after all, but this has been realized more recently. A good example of that is that there is a locale $X$ with no points, and an injective map $ X \to \mathbb{R}$ which is the natural domain of definition of functions that are defined "almost everywhere": as such functions cannot be evaluated at any specific point, this locale cannot have any point. I think this example is studied in length in Alex Simpson Measures, Randomness and sublocales (or maybe what he is studying is the image of this locale in $\mathbb{R}$).

6. The main interest of locale theory is to realise that topology doesn't really care about points, and to some extent works better if one does not care about points. So trying to construct faithful functor from the category of locales to the category of sets is a little bit weird from this perspective. I could have answered to your comment " It is intuitive that continuous maps map points to points" that it is also the case with locales that points are sent to points, but it's just that we don't really care about points.

In fact, before Cantorian set theory which pushed everyone in mathematics to think about everything as a set, most mathematicians were not thinking about spaces like the real numbers as being sets + a topology (in the Cantor style view of sets as a discrete objects) but really as a "continuum" not formed of "discrete points". This is the point of view that locale theory is pushing forward. But if you really want to have a faithful functor from locales to sets, which has some geometrical meaning, here is a way to get one:

If you start from a frame $A$ you can see it as a distributive lattice and attached to it its Stone spectrum Stone(A) which is the compact (in general not Hausdorff) space of prime filters of $A$. Morphisms of frames are a special case of morphisms of distributive lattices, so this produces your faithful functor and I think it is the most geometric one can come up with.

The points of the corresponding locales $Loc(A)$ are the totally prime filters of $A$: "prime filter" means if $a \cup b \in P$ then $a \in P$ or $b \in P$, while "totally prime filter" means if $ \bigcup a_i \in P$ then $\exists i, a_i \in P$. So the points of the locale form in some sense a specific subspace of $Stone(A)$.

From the localic point of view this can be promoted to a morphism of locales $Loc(A) \rightarrow Stone(A)$, and it is always the case that the locale $Loc(A)$ is a dense subspace of $Stone(A)$, but of course $Loc(A)$ can have no points. You can think of points of $Stone(A)$ as an approximation to (eventually non existing) points of $Loc(A)$. But I am not sure this picture gives the correct insight on locales: for example if you applied this to the locale corresponding to an ordinary topological space (like the real number) you will get a very big and not very natural space $Stone(A)$ which is a lot more complicated than the space you started with.

Answer 2

While Simon's answer is very good, I think one can also say something a little more along the lines of what you may be thinking. I haven't seen this written out in this way before, so I may have made some errors, but I think the general idea is valid.

Note that the "space of points of a locale", when regarded as a functor on frames, is very similar to the maximal or prime spectrum of a ring. For a ring $R$, the points of $\mathrm{Spec}(R)$ are the prime ideals in $R$, and the open sets are generated by the sets $O_f$ of prime ideals not containing $f$ for $f\in R$. Similarly, for a frame $A$, the points of "$\mathrm{Spec}(A)$" are the completely prime filters in $A$ (filters and ideals being complementary in a lattice), and its open sets are generated by sets $O_u$ of filters containing $u$ for $u\in A$.

Now, as we know, a ring $R$ induces a sheaf of rings on $\mathrm{Spec}(R)$, whose ring of sections over the open set $O_f$ generated by some $f\in R$ is the localization $R_f$. Similarly, from a frame $A$ we can try to induce a sheaf of frames on $\mathrm{Spec}(A)$, whose frame of sections over $O_u$ is the set $A_u$ of elements of $A$ that are $\le u$; this is the "localization" at $u$ in frames, since the only "unit" in a frame is the top element.

The structure sheaf of a ring $R$ is a sheaf of local rings, in the sense that its stalks are local rings. A ring is local if $\sum_{i=1}^n a_i = 1$ implies there exists an $a_i$ that is invertible. Similarly, a frame is local if $\bigvee_i a_i = \top$ implies there exists an $a_i=\top$, i.e. the set $\{\top\}$ is a completely prime filter, or equivalently (in localic language) there is a point whose only neighborhood is the whole space (a "focal point"). The above sheaf of frames would be a local one; indeed, its stalk over a point $x\in \mathrm{Spec}(A)$ is the "localization of $A$ at $x$" in an appropriate sense (localically, it is the "germ at $x$", the intersection of all the open sublocales containing $x$).

The problem, of course, is that unlike rings, frames may not have enough points. One manifestation of this is that a localization of a ring $R_f$ depends only on the radical of the principal ideal $(f)$, and hence only on the set of prime ideals containing (or not containing) $f$; thus the definition $O_{\mathrm{Spec(R)}}(O_f) = R_f$ is well-defined in that if $O_f = O_g$ then $R_f \cong R_g$. However, this is not true for frames: we can have $O_u=O_v$ but $A_u\ncong A_v$. The most drastic example is that there are nontrivial frames without any points at all, so that $O_u=O_v$ for any $u,v$. This means that our above attempted definition of a "structure sheaf of frames" on $\mathrm{Spec}(A)$ doesn't work.

We can, I believe, fix it by taking colimits of $A_u$ over all appropriate $u$, and the stalks will still be as I claimed above. But the problem remains that the space $\mathrm{Spec}(A)$ itself could be completely empty even if $A$ is highly nontrivial, and so frames will not embed contravariantly into "locally framed spaces" in the same way that rings embed into locally ringed spaces.

The solution, of course, is that we need to consider locally framed locales rather than spaces. Since this is obvious to locale theorists, they don't generally go through the above detour. But now the analogous construction looks very tautological: the locale version of the spectrum $\mathrm{Spec}_\ell(A)$ is just $A$ itself regarded as a locale. The "structure sheaf" is the subobject classifier of the sheaf topos $\mathrm{Sh}(A)$, which as is well-known is an internal frame, and the colimit-over-$u$ fix I described above is the pullback of this internal frame to the topos of sheaves over the topological space of points $\mathrm{Spec}(A)$.